Chances solutions   

Solutions by Richard Cowan

An important note: The problems are not stated in their entirety on this page, only in a brief form to jog one's memory. Please refer to the CHANCES front page or archives for the full background information of each problem.

Problem 1 (from the club's August-September, 2001 Bulletin)

West leads 5 against South's 4S. East wins with A, returns 7 and ruffs West's continuation. South overruffs and draws trumps in two rounds. Now ...

  • Play the A first, claiming if Q falls or hearts break 0-7 (whence ducking a club to East gives a marked endplay).
  • Otherwise lead a club toward dummy intending to play the 10, but covering any honour from West (in which case the remaining clubs yield the needed 3 tricks). Should West show out on the first round of clubs, rise with the A and claim (citing the line "K then J").
  • If the 10 loses to East, win the heart return.
  • If Q falls, claim. If West shows out, repeat the club finesse losing only if East has QJ.
  • Otherwise play the A. If the other club honour doesn't drop, cash 2 spades pitching South's 3 and retaining the lead with North.
  • The contract now makes if clubs were 3-3 originally or on a squeeze if East had the Q and 4+ clubs initially. (Note: To keep entries fluid, one must have played trumps carefully, leaving the second highest outstanding spade with South.)

   North
 KJ74
 J62
 QJT
 AT9

  South
 AQT3
 AK3
 84
 K432

Overall chance = 87.25%.   Details available (as Word file); 5 competing lines are analysed.

Problem 2 (from the club's October-November, 2001 Bulletin)

South, in 6NT, receives the J lead. Now ...      Play to the K and run the J.  
If this wins: Lead low to 10 and (provided nobody has 5+ cards) cash the last heart. Cash the Q (claiming if East shows out, citing a ducked spade into East). Otherwise play spades, aiming for 3 tricks there. Low to A, claiming if West plays an honour. Run the 9 next.

  • If East covers win and claim.
  • If East shows out, play K and finesse the J hoping to make 3 diamond tricks by either a well-placed Q or a 3-3 split.
  • If the 9 loses to West, win his club exit, cash any remaining heart and the K, claiming if spades are 3-3. If not, play the K and finesse the J as one's last chance.

If the J loses to West:  Win the club exit, cash the hearts, then the K and the diamond finesse which we need. If the A reveals a 3-3 break, we win, but if not we are in dummy with the position as shown.
          North:  A9      - 5 A
          South:  K1083- - -
If West, who has the Q, has 4+ spades or QJx, he is squeezed. Even if West follows to the A, should he now play an honour under the A, we win (as counting will reveal whether it is a singleton or QJ bare).

   North
 A97
 KJ2
 AJ52
 A86

  South
 KT83
 AT95
 K64
 KQ

Overall chance = 60.02%.   Details available (as pdf file); 3 basic lines are analysed.

Problem 3 (from the club's "December2001-January 2002" Bulletin)

West leads a trump against 6C. East follows and South wins with the A. On the Q, both follow. Now ...

One line is based on playing off the 6 outside top cards and then, if no favourable drop has assured 12 tricks, leading low to the J8, planning to ruff a heart if necessary. This appealed to many and it is almost the best at 87.98%. The best line is a tad better.

Play to the K, back to J and then low toward the A107, inserting the 10 if West follows low (claiming if the 10 holds). Play the A instead, and claim, if West shows out or plays an honour. Otherwise, win East's diamond exit with the A (pitching 8) and claiming if the 10 is high. If not, cash A and K; claim if the Q has fallen. If it hasn't, play A (claiming if the remaining honour falls or West shows out). If not, play a club to reach the following position.
       North: 7    J -  10
       South: K9 - 10 -
There is a chance that East may already have been squeezed in three suits (if he were dealt 4+ spades, both red queens and the J). If not, play the 10. East will now be subject to a - squeeze if dealt 4+ spades and both diamond honours. If he discards the master diamond, claim. If not, pitch the 10 and rely on either the - squeeze (against either defender with 4+ spades and the Q) or the prosaic 3-3 spade break.

   North
 AT7
 AKJ8
 K
 KT942

  South
 K962
 753
 AT2
 AQJ

Overall chance = 89.35%. Details available (as pdf file).

Problem 4 (from the club's "February-March, 2002" Bulletin)

The J is led against South's 2NT. East wins with the K and leads the 2; West's A captures the Q. West plays the 10 and 8 (the suit splitting 4-4 predictably). West exits with the 2. Now ...

The winning line is simplicity plus. Discard the 3 and 2 from dummy and the 4 from hand. Win the heart exit with the K and play the A.

  • If an honour falls -- indicative of a singleton honour or KQ bare -- play to the J to establish a second diamond trick. Now declarer needs only 3 club tricks and can achieve this by taking a club finesse or two when in dummy (assuming that East has the K). Club-suit splits and the location of the 9 are irrelevant.
  • If both defenders follow low on the A, success depends on bringing home 4 club tricks. Cross to the A and run the J. If it is covered, win and cross to the 10, cash the Q and come back to the Q, hoping for a 3-3 break or the fall of the 9. If the J holds, cash Q and run the 10. If that is covered. win and play off the clubs.

Lines based on a double finesse in diamonds are a tad worse.

   North
 97
 AQ73
 JT62
 JT8

  South
 Q65
 K4
 A974
 AQ73

Overall chance = 36.4%. Details available as a pdf file.

Problem 5 (from the club's "April-May, 2002" Bulletin)

West leads the Q against 4H; East wins and returns the Q to West's 7. Now ...

Everyone won the spade, but then personality came in. People who save for their old age were attracted to the line of ruffing a club, then drawing 3 rounds of trumps ending in dummy. If clubs are no worse than 4-2 and trumps break 3-2, one claims. If not, there are still chances if the long trumps and long clubs are in the same hand. The overall chance is 66%.

The "spenders" cashed high clubs early, hoping to pitch spades. If over-ruffed, they then fell back on either good cross-ruff chances (often exploiting newly-established spade shortage) or further club tricks -- depending on circumstances.

Here the early-spending line is far superior. Any early spade pitches achieved are like gold. Moreover, the line does not focus solely upon that marvellous club resource; it recognizes the other fine resource, namely the excellent trump pips.

   North
 A92
 AQ9
 K
 AKQ953

  South
 865
 KT876
 6542
 7

Overall chance of best line = 89.45%. Full solution available as a pdf file.

Problem 6 (from the club's "June-July, 2002" Bulletin)

West leads the 8 against 6NT; East wins and returns the 4 to the T, 9 and K. The finesse wins. Now ...

- - - AK97

- - T  Q82

 Cross to the A and repeat the finesse. Play the A; claim if the K has fallen. If not, play off all major-suit winners ending in hand and reaching the end-position shown. Observe discards carefully.

East is squeezed if he has 4+ clubs and long diamonds, so the only danger comes when West has 4+ clubs. If South knew that West had exactly 4 clubs, he might reason (correctly) that the chance of her having JTxx is 60%, with Jxxx and Txxx accounting for 20% each. So he might opt for the seemingly better play of  low to the 9, covering if West splits honours (whereby the Q provides an entry for a repeat finesse), 60% compared to only 40% using the alternative, low to the A.
   It is not straightforward, though, because West can discard in a manner to give or withhold information as to her precise shape. For example, if West has 5-3-1-4, she can pitch a and a to tell South that she is 5-3-1-4 exactly (and perhaps induce a "low to the 9" play in clubs, losing to a singleton honour) or pitch two s to keep South guessing as to 5-3-1-4 or 5-4-1-3 (the and lengths being revealed in the course of play). This is a "sub-game" of the type seen in mathematical books on "game theory"; West has choices of discard and South has choices between the "drop" and "double finesse". There are a number of other "poker-like" games of bluff and counter-action in this seemingly benign deal..



   North
 AK4
 KQJ
 64
 AK973

  South
 QJ9
 T6
 AQJT2
 Q82

The full solution, with details of these "game-theoretic issues"  is available as a pdf file.
The overall chance using the best line = 91.46%.

Problem 7 (from the club's "Aug-Sept, 2002" Bulletin)

West leads the Q against 4H after the auction 1H-2H <> 3C-4H. How best to play 4H by South?

After winning trick 1, there are two basic approaches -- drawing trumps and hoping for 4 tricks in the minor suits, or playing a dummy reversal and hoping for 3 minor-suit tricks.

The former comes in at around 44% (calculated with due allowance for inferences based on the silence of the opponents in the bidding). The latter, which involves ruffing 3 spades high in the South hand with low trumps led to enter dummy, is much better.

One issue (discussed in the details) is the appropriate continuation on trick 7. Although "low to the K" appeals as a quick way to get to dummy and play the boss trump,  it is better to play "low to the  J". This advice holds regardless of the heart and spade breaks, although both approaches are almost equal in merit in the most benign case where hearts are 3-2 or 2-3 and spades are 4-4.

   North
A876
 JT9
 842
 KJ7

  South
5
 AKQ62
 K75
 A643

 

Overall chance using the best line = 70.48%. Details available in the pdf file.

Problem 8 (from the club's "Oct-Nov, 2002" Bulletin). 

Was Victoria's 7D the correct bid or  was the less ambitious 6D of NSW the more appropriate contract in their head-to-head match at the 2002 ANC Interstate Teams? 

7D is better.  At 79.05%, using the line of ruffing spades in hand, the chance for 7D exceeds the conservative threshold of 66% (calculated assuming an absolutely certain 6D contract).  The actual threshold here, where 6D is not cast iron, is slightly lower than 66%.

We win the opening trick (with the A if a club lead) and play the A.
·   If West shows out, concede.
·   If East shows out, cross to hand with the A and play the 10. It is better for West to cover. Win in dummy and  return to the 9. Play to the K and ruff a spade.  Hope that spades either break 3-3 or the Q falls single or double from East.
·   If both follow to the A,  play the K. Claim if trumps are 2-2. If not, the strategy depends on whether or not the J has fallen. Details are contained in the downloadable pdf file.

   North
KJ972
 9
 AKQ32
 AJ

  South
A5
 A653
 T964
 KQ8

Overall chance for 7D using the best line = 79.05%. Details available in the pdf file.

Problem 9 (from the club's "Dec, 2002-Jan, 2003" Bulletin). 

1.    a) How should one play the trump suit KQ643 opposite dummy’s 1087 for one loser?  (b) What is the best line and by how much does one’s chance decline from (a) if the 7 is used first for ruffing?

2.    In 3NT with other suits well stopped, North has J10732 and South A6, needing 3 tricks in the suit.  How to play when the defence knows that South doesn’t have 3-card support for spades?

3.   (a) How should declarer play K982 (in dummy) opposite A54 for 3 tricks?  (b) What about shuffling the pips: K952 and A84?

4.   How best to play the trump suit – A6 in dummy and Q98542 in hand, for at most one loser?

See the pdf file for the solution

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